mapped weighted essentially non-oscillatory schemes ...powers/paper.list/jcp.weno5m.05.pdf ·...

Journal of Computational Physics 207 (2005) 542–567

www.elsevier.com/locate/jcp

Mapped weighted essentially non-oscillatoryschemes: Achieving optimal order near critical points q

Andrew K. Henrick a, Tariq D. Aslam b,*, Joseph M. Powers a

a Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556-5637, USAb MS P952, Dynamic Experimentation Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 10 September 2004; received in revised form 28 January 2005; accepted 28 January 2005

Available online 17 March 2005

Abstract

In this paper, a new fifth-order weighted essentially non-oscillatory scheme is developed. Necessary and sufficient

conditions on the weights for fifth-order convergence are derived; one more condition than previously published is

found. A detailed analysis reveals that the version of this scheme implemented by Jiang and Shu [G.-S. Jiang, C.-W.

Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] is, in general, only

third-order accurate at critical points. This result is verified in a simple example. The magnitude of �, a parameter which

keeps the weights bounded, and the level of grid resolution are shown to determine the order of the scheme in a non-

trivial way. A simple modification of the original scheme is found to be sufficient to give optimal order convergence even

near critical points. This is demonstrated using the one-dimensional linear advection equation. Also, four examples uti-

lizing the compressible Euler equations are used to demonstrate the scheme�s improved behavior for practical shock

capturing problems.

� 2005 Published by Elsevier Inc.

Keywords: WENO; Hyperbolic equations

1. Introduction

Weighted essentially non-oscillatory (WENO) finite difference schemes have become one of the most

popular methods for approximating solutions to hyperbolic conservation equations. In this paper, the

fifth-order WENO method is analyzed using a simple one-dimensional model problem given by

0021-9991/$ - see front matter � 2005 Published by Elsevier Inc.

doi:10.1016/j.jcp.2005.01.023

q This study has been supported by Los Alamos National Laboratory and performed under the auspices of the US Department of

Energy.* Corresponding author. Tel.: +1 505 667 1367; fax: +1 505 667 6372.

E-mail addresses: [email protected] (A.K. Henrick), [email protected] (T.D. Aslam), [email protected] (J.M. Powers).

mailto:[email protected]



A.K. Henrick et al. / Journal of Computational Physics 207 (2005) 542–567 543

ouot

þ ofox

¼ 0; ð1Þ

where u(x, t) is a conserved quantity, f(u(x, t)) describes its flux, and x and t denote space and time, respec-

tively. In general, the domain t 2 [0, 1) and x 2 [x0, xN] is considered. The results found by examination of

this model problem can be generalized for systems of higher dimension.

Hyperbolic problems characterized by Eq. (1) admit discontinuous solutions. Such discontinuities intro-

duce spurious oscillations in the approximate solutions generated by many traditional numerical methods.

In order to address this issue, total-variation diminishing (TVD) schemes were developed in [2], founded onthe work of Van Leer [3,4]. Later, Harten et al. [5] introduced essentially non-oscillatory (ENO) schemes.

Instead of using a single fixed stencil to approximate spatial fluxes, an rth order ENO scheme uses a set of r

candidate stencils. An indicator of smoothness of the solution is determined over each stencil. The stencil

over which the solution is smoothest is then chosen in an attempt to diminish the effect of a discontinuity on

neighboring cells. This method was further developed by Shu and Osher [6,7] who introduced pointwise

ENO schemes which involve significantly fewer computations for multidimensional problems with inhomo-

geneous source terms than their cell-averaged counterparts.

Subsequently, Liu et al. [8] introduced WENO schemes, again using a cell-averaged approach. In suchschemes, spatial derivatives are calculated using a convex combination of the numerical fluxes associated

with each candidate stencil. This is accomplished by weighting the contribution of each flux according

to the smoothness of the solution over each stencil. Using the indicator of smoothness introduced by

Liu et al., an rth order ENO scheme can be converted into an (r + 1)th order WENO scheme. Suggesting

a new indicator of smoothness, Jiang and Shu [1] then developed WENO5, a pointwise WENO scheme.

WENO5 converges at fifth order in special cases and is differentiated from other fifth-order WENO

schemes by the weights used in [1]. Balsara and Shu [9] showed that a (2r � 1)th order WENO scheme

can be constructed from the stencils of an rth order ENO scheme for r = 4, 5, 6 and 7. Also, Fedkiwand co-workers [10,11] have noted that �, a parameter in these schemes which introduces a numerical bound

on the weights, is a dimensional quantity and have suggested an appropriate scaling.

Despite the fifth-order convergence behavior often exhibited by WENO5, its actual rate of convergence

is less than fifth order for many problems. In fact, the analysis done in [1] does not identify all the properties

of the weights which are necessary for a WENO scheme to converge at fifth-order. Furthermore, the incom-

plete set of properties given in [1] is not satisfied by the WENO5 weights when the first derivative of the

solution vanishes. In particular, at critical points where the third derivative does not simultaneously vanish

with the first, WENO5 suffers a loss in accuracy. This formal loss of optimal fifth-order convergence ratehas not been previously recognized in the literature. Because solutions will often involve critical points

where the first derivative vanishes while the third derivative does not, this behavior is an important issue

for many problems.

In this paper, a refinement of the scheme of [1] is developed which has formal fifth-order convergence

properties. Using the weights as formulated in [1] as a starting point, the modified scheme renders these

weights closer to optimality so as to satisfy sufficient criteria for fifth-order convergence. The resulting

scheme maintains its fifth-order accuracy even when � is chosen to be the smallest number possible for a

given machine precision. Furthermore, the effects of the parameter � on previous WENO schemes areisolated and examined. Notably, the convergence properties of such schemes are problem-dependent and

cannot be known a priori. A detailed derivation of the WENO scheme is given. Section 2 clarifies the con-

cepts of numerical and actual flux, especially as they pertain to the formulation of spatial derivatives in con-

servation equations. Following the presentation given in [1], the coefficients for the fifth-order upwind

scheme, commonly referred to in the literature as the upstream central scheme, are derived. In Section 3,

standard concepts behind WENO schemes are elucidated. The candidate stencils for the fifth-order WENO

scheme are developed as well as the weights for combining the candidate stencils such that the upstream

544 A.K. Henrick et al. / Journal of Computational Physics 207 (2005) 542–567

central scheme is recovered. Necessary and sufficient conditions on the non-oscillatory weights to achieve

fifth-order convergence are derived. Section 4 gives the form of the non-oscillatory weights used in [1].

These weights are shown to violate the necessary conditions for fifth-order convergence near critical points.

Furthermore, the order of the scheme is shown to be strongly sensitive to the parameter �. In Section 5, a

modification is proposed which formally gives fifth-order convergence at these points. This modificationinvolves mapping the weights as calculated according to [1] to values such that the necessary optimal con-

ditions on the weights are satisfied. As shown for several examples, the new mapped weights retain the ENO

behavior. Section 6 addresses the corrections necessary for the WENO schemes up to 11th order.

Several challenging examples of flows with discontinuities are given. In each case, side-by-side compar-

isons of the predictions of the improved algorithm relative to the original algorithm of [1] show the

improvement gives rise to more accurate predictions; however, relative to smooth flows, the accuracy gains

are more modest. The achieved convergence rate, determined for the Sod [12] shock tube problem, is

roughly first order; a result such as this is not unexpected when any shock capturing scheme is appliedto a flow with a discontinuity.

2. Conservation, flux, and numerical flux

A conservative numerical scheme ensures that a quantity remains conserved by calculating a single flux

which describes the flow of that quantity between neighboring cells. Though flux conservative schemes

are normally formulated using finite volumes, a finite difference scheme is developed here using an approachsimilar to Shu et al. [6]. In the case that Eq. (1) involves an inhomogeneous source term, these finite difference

schemes have the computational advantage of removing the need to integrate such a term over each cell [6].

In order to formulate this finite difference scheme, the discretization shown in Fig. 1 is used. Indices are

assigned using the same label for each cell and corresponding node; the position of node j is denoted by xj,

where j = 0,1, . . ., Nx, corresponding to the domain introduced for Eq. (1). Similarly functional values at

node j will be denoted by a subscript j (e.g., fj = f(xj)). Half indices indicate the edges of cells across which

fluxes will be calculated. Uniform grid spacing is maintained such that xj = x0 + jDx, where Dx ¼ xNx�x0Nx

.

Eq. (1) can be approximated by a system of ordinary differential equations, where the spatial derivativehas been replaced by a finite difference to yield:

dujdt

¼ � fjþ1=2 � fj�1=2

Dx: ð2Þ

Such a method is conservative since a single flux function is used to describe the flow of a conservativequantity between two cells: fj + 1/2 describes the flux of u between cells j and j + 1.

2.1. The numerical flux function

Higher orders of accuracy can be achieved without sacrificing the conservative character of the scheme

by considering the form of Eq. (2). Even if f is inconsistent with the actual flux function, u is conserved in an

node- cell boundary-

xj+1/2j-1/2

jj-2 j-1 j+1 j+2j-3 Nx 0

Fig. 1. Computational grid.


infinite domain due to the telescoping cancellation of fluxes between cells. By substituting a different func-

tion for f in Eq. (2), the accuracy of this conservative scheme can be increased. Of course any replacement

flux function must be chosen such that solutions generated by Eq. (2) approximate in a consistent manner

those of the original partial differential equation, Eq. (1).

A function is sought for which the divided difference in Eq. (2) is the exact derivative at xj rather than onlyan approximation. Such a function is known in the literature as the numerical flux function, h(x), and satisfies

ofox

��x¼xj

¼ hjþ1=2 � hj�1=2

Dx: ð3Þ

Following Shu and Osher [7], h(x) is defined according to

f ðxÞ ¼ 1

Dx

Z xþDx=2

x�Dx=2hðnÞ dn: ð4Þ

Differentiation of Eq. (4) and evaluation at x = xj shows that h(x) satisfies Eq. (3) exactly.

Eq. (4) defines h(x) implicitly such that

dujdt

¼ � hjþ1=2 � hj�1=2

Dxð5Þ

involves no error due to approximating the spatial derivative with a finite difference. Thus h(x) is selected as

a replacement for the actual flux in Eq. (2). While h(x) = f(x) + O(Dx2), as shown in [11], it is clear that h(x)

is not the exact flux for a fixed finite Dx.

2.2. Higher order spatial derivatives

By approximating h(x) in Eq. (5), conservative numerical schemes are formulated. These approximations

of h(x) are denoted by f̂ ðxÞ and are constructed using a polynomial form with undetermined coefficients.Substitution of this polynomial into Eq. (4) leads to a system of equations where the flux is a known quan-

tity at the nodes surrounding the interface of interest, allowing for a unique set of coefficients to be found.

Having found f̂ , the spatial derivative in Eq. (1) is approximated by

ofox

��x¼xj

� �f̂ jþ1=2 � f̂ j�1=2

Dx: ð6Þ

Two orders of convergence will be of concern. Clearly the order at which Eq. (6) is satisfied is of primaryimportance since it is the order at which the overall scheme converges in space. Also the order of individual

approximations to the numerical flux, f̂ j�1=2, are important, since they are used to develop the criteria for

admissibility of non-oscillatory weights.

2.3. The upstream central scheme

The upstream central scheme is derived by considering a fourth degree polynomial approximation

hðxÞ � f̂ ðxÞ ¼ a0 þ a1xþ a2x2 þ a3x3 þ a4x4; ð7Þ
with undetermined coefficients, ak, where k = 0, . . ., 4. Substituting Eq. (7) into Eq. (4) and performing the
integration gives

f ðxÞ ¼ a0 þ a1xþ a2 x2 þ Dx2

12

� �þ a3 x3 þ Dx2x

4

� �þ a4 x4 þ Dx2x2

2

� �: ð8Þ

x

f j+1/2

i=j-3 i=j-2 i=j-1 i=j i=j+1 i=j+2i=0 i=Nx

x xxx xj-2 j-1 j j+1 j+2

Fig. 2. Upstream central scheme used to compute f̂ jþ1=2.


Referring to Fig. 2, the ak are found by evaluating Eq. (8) at the stencil nodes, resulting in a system of fiveequations in the five unknown coefficients. As seen in Fig. 2, the stencil is centered around xj by using the

nodes xi = xj + (i � j)Dx for i = j � 2, j � 1, j, j + 1, j + 2. Accordingly, the ak are found in terms of the

nodal fluxes, xj, and Dx.Substituting these values for the ak into Eq. (8) and evaluating the numerical flux at the cell interface

xj + 1/2 gives

f̂ jþ1=2 ¼ 160

2f j�2 � 13f j�1 þ 47f j þ 27f jþ1 � 3f jþ2

� �¼ hjþ1=2 �

1

60

d5fdx5

��x¼xj

Dx5 þOðDx6Þð9Þ

by Taylor series expansion. Note that both xj and Dx are not present in the leading order term in Eq. (9);thus, it is clear that the resulting stencil is independent of both and may be used to evaluate the numerical

flux at any interface in the domain.

To calculate the numerical flux at x = xj � 1/2, the stencil is shifted to the left by one grid spacing to give

f̂ j�1=2 ¼ 160

2f j�3 � 13f j�2 þ 47f j�1 þ 27f j � 3f jþ1

� �¼ hj�1=2 �

1

60

d5fdx5

��x¼xj

Dx5 þOðDx6Þð10Þ

using another Taylor series expansion. While f̂ j�1=2 only approximates hj±1/2 to O(Dx5), the Dx5 error termis the same for both stencils.

Substitution of Eqs. (9) and (10) into Eq. (6) gives

dfdx

��xj

� 1

60�2f j�3 þ 15f j�2 � 60f j�1 þ 20f j þ 30f jþ1 � 3f jþ2

� �¼ f 0 þOðDx5Þ:

ð11Þ

Despite the fact that f̂ is only fifth-order accurate and that Eq. (6) involves division by Dx, a fifth-order

approximation of the derivative is retained. As a result of using the same stencil to approximate

f̂ jþ1=2 and f̂ j�1=2, the lowest order truncation error terms are equal, as noted, and cancel.

3. Fifth-order WENO schemes

In order for this numerical method to better approximate derivatives near shocks, problems associated

with using a large stencil, such as that shown in Fig. 2, must be addressed. In the case of such a large stencil,


a discontinuous solution will affect five different nodes at once and will cause the numerical solution to spu-

riously oscillate. Thus, a method is sought which smoothly changes the stencil in the neighborhood of a

discontinuity in order to avoid these problems. Fifth-order WENO schemes are devised to implement this

solution technique [1].

Fifth-order WENO schemes are based on the stencils shown in Fig. 3. A numerical flux,f̂k

jþ1=2 ¼ hjþ1=2 þOðDx3Þ with k 2 f0; 1; 2g, is calculated for each of the three point stencils shown. The

f̂k

jþ1=2 from these stencils are then combined in a weighted average such that in smooth regions this method

mimics the central scheme given in Eq. (9). In regions of the flow which contain discontinuities, weights are

assigned such that the solution is essentially non-oscillatory.

The numerical flux is now calculated according to

hjþ1=2 � f̂ jþ1=2 ¼X2k¼0

xk f̂k

jþ1=2; ð12Þ

where xk is the weight corresponding to stencil k.

3.1. Formulation of the f̂k

jþ1=2

Each of the f̂kðxÞ calculated gives a third-order approximation of h(x). This is done by the following the

same procedure given in Section 2.1. A polynomial form is postulated

hðxÞ � f̂kðxÞ ¼ b0 þ b1xþ b2x2; ð13Þ

and b0, b1 and b2 are found according to Eq. (4). Thus, for each stencil shown, a different third-order

approximation for the numerical flux function is found.

The numerical flux functions for each stencil are found to be

f̂0ðxÞ ¼

�fj�2 þ 2f j�1 þ 23f j

24þ

fj�2 � 4f j�1 þ 3f j

2Dx

� �xþ

fj�2 � 2f j�1 þ fj2Dx2

� �x2;

f̂1ðxÞ ¼

�fj�1 þ 26f j � fjþ1

24þ fjþ1 � fj�1

2Dx

� �xþ

fj�1 � 2f j þ fjþ1

2Dx2

� �x2;

f̂2ðxÞ ¼

23f j þ 2f jþ1 � fjþ2

24þ

�3f j þ 4f jþ1 � fjþ2

2Dx

� �xþ

fj � 2f jþ1 þ fjþ2

2Dx2

� �x2;

ð14Þ

x

Stencil 2

Stencil 1

j j+1

Stencil 0

j+1/2f

Fig. 3. Stencils used for WENO5 numerical flux.


where the stencils are centered around xj = 0. It is noted that b1 and 2b2 for each equation are finite differ-

ence approximations of f 0j or f 00

j , respectively.

These approximations are evaluated at the j + 1/2 interface to give the f̂k

jþ1=2 of Eq. (12):

f̂0

jþ1=2 ¼ 16ð2f j�2 � 7f j�1 þ 11f jÞ;

f̂1

jþ1=2 ¼ 16ð�fj�1 þ 5f j þ 2f jþ1Þ;

f̂2

jþ1=2 ¼ 16ð2f j þ 5f jþ1 � fjþ2Þ:

ð15Þ

Since these linear combinations define the numerical flux at every cell boundary in the domain, one needs

only to shift each index by �1 to arrive at the corresponding f̂k

j�1=2.

The Taylor series expansions of Eq. (15) give

f̂0

j�1=2 ¼ hj�1=2 � 14f 000ð0ÞDx3 þOðDx4Þ;

f̂1

j�1=2 ¼ hj�1=2 þ 112f 000ð0ÞDx3 þOðDx4Þ;

f̂2

j�1=2 ¼ hj�1=2 � 112f 000ð0ÞDx3 þOðDx4Þ:

ð16Þ

Note that each of these takes the form

f̂k

j�1=2 ¼ hj�1=2 þ AkDx3 þOðDx4Þ; k 2 f0; 1; 2g: ð17Þ

Here the third-order error term has been explicitly written in Eq. (16) since it will become important in

deriving constraints on the non-oscillatory weights.

Note also from Eq. (16) that

1

Dxf̂n

jþ1=2 � f̂m

j�1=2

� �¼

ofox

��x¼xj

þOðDx2Þ if n 6¼ m;

ofox

��x¼xj

þOðDx3Þ if n ¼ m:

8<: ð18Þ

3.2. Ideal weights

In regions of smooth flow, the linear combination of the f̂k

jþ1=2 given in Eq. (12) should reduce to the

central scheme, since the central scheme gives the optimal convergence properties possible for a five point

stencil (cf. Figs. 2 and 3). These weights are thus known in the literature as the ideal weights. Substituting

Eq. (15) into Eq. (12) gives the numerical flux for fifth-order WENO schemes. Setting this numerical fluxequal to the upstream central scheme given in Eq. (9) gives

f̂ jþ1=2 ¼1

6

2f j�2 � 7f j�1 þ 11f j

�fj�1 þ 5f j þ 2f jþ1

2f j þ 5f jþ1 � fjþ2

0B@

1CA

T

��x0

�x1

�x2

0B@

1CA ð19Þ

¼ 1

602f j�2 � 13f j�1 þ 47f j þ 27f jþ1 � 3f jþ2

� �;

where �xi is the ith ideal weight. These weights, which satisfy

X2k¼0

�xk f̂k

jþ1=2 ¼ hjþ1=2 þOðDx5Þ ð20Þ


are given by

�x0 ¼ 1=10; �x1 ¼ 6=10 and �x2 ¼ 3=10: ð21Þ
Note that they sum to unity. Furthermore, use of this stencil results in f 0ðxÞ ¼ hjþ1=2�hj�1=2
Dx þOðDx5Þ, as seenin Section 2.2.

3.3. Non-oscillatory weights

In order to achieve the optimal order approximation to hj + 1/2 in smooth regions, the weights must con-

verge in an appropriate manner to the ideal weights as Dx approaches zero; however, in regions where a

discontinuity does exist, the weights should effectively remove the contribution of stencils which contain

the discontinuity.In order to arrive at appropriate criteria for the weights, consider that the numerical flux function is

given by

f̂ j�1=2 ¼X2k¼0

xkf̂k

j�1=2; ð22Þ

where f̂k

j�1=2 ¼ hj�1=2 þ AkDx3 þOðDx4Þ. Adding and subtractingP2

k¼0 �xk f̂k

j�1=2 from Eq. (22) and compar-ing with Eq. (20) gives

f̂ j�1=2 ¼X2

k¼0�xk f̂

k

j�1=2|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}hj�1=2þOðDx5Þ

þX2k¼0

ðxk � �xkÞf̂k

j�1=2; ð23Þ

where the second term must be at least an O(Dx6) quantity in order for the derivative to be approximated at

fifth order.Expanding this second term gives
X2
k¼0

ðxk � �xkÞf̂k

j�1=2 ¼X2k¼0

ðxk � �xkÞðhj�1=2 þ AkDx3 þOðDx4ÞÞ

¼ hj�1=2

X2k¼0

ðxk � �xkÞ þ Dx3X2k¼0

Akðxk � �xkÞ þX2k¼0

ðxk � �xkÞOðDx4Þ: ð24Þ

Thus, it is sufficient to require
X2k¼0
ðxk � �xkÞ ¼ OðDx6Þ ð25aÞ

and

xk � �xk ¼ OðDx3Þ ð25bÞ
for the overall scheme to retain fifth-order accuracy. Note that, while these conditions are not necessary,
they serve as a simple set of criteria around which to design the non-oscillatory weights.

Thus far, the accuracy requirements have been found by considering each individual numerical flux in

isolation; however, Eq. (6) reveals that it is the difference in the numerical fluxes across a cell which actuallydetermines the accuracy of the scheme. In considering f̂ jþ1=2 � f̂ j�1=2 ¼ f 0ðxÞDxþOðDx6Þ, it will be shownthat Eq. (25b) may be relaxed.

Employing the same technique using in deriving Eq. (23), comparison of the ideal weights with the non-

oscillatory weights conveniently identifies the error terms. In this case, however, the constraints are found

by examination of the the error terms which result from subtraction of the numerical fluxes:


f̂ jþ1=2 � f̂ j�1=2 ¼X2

k¼0�xk f̂

k

jþ1=2 �X2

k¼0�xkf̂

k

j�1=2

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{f 0jþOðDx6Þ

þX2k¼0

ðxðþÞk � �xkÞf̂

k

jþ1=2 �X2k¼0

ðxð�Þk � �xkÞf̂

k

j�1=2;

ð26Þ
where superscript (+) or (�) on xk indicate their use in either f̂ jþ1=2 or f̂ j�1=2 stencil. Expanding the last two
terms in Eq. (26) and substituting for f̂k

j�1=2 from Eq. (16), the criteria for fifth-order convergence are found

to be

X2k¼0

ðxð�Þk � �xkÞ ¼ OðDx6Þ; ð27aÞ

X2k¼0

AkðxðþÞk � xð�Þ

k Þ ¼ OðDx3Þ ð27bÞ

and

xð�Þk � �xk ¼ OðDx2Þ: ð27cÞ

Carrying out the sum in Eq. (27b) yields

� 1

123xðþÞ

1 � 3xð�Þ1 � xðþÞ

2 þ xð�Þ2 þ xðþÞ

3 � xð�Þ3

� �f 000ð0Þ ¼ OðDx3Þ ð28Þ

taking into account that the Ak are given explicitly in Eq. (16).

Thus to retain fifth-order accuracy, the necessary and sufficient conditions are

X2k¼0

ðxk � �xkÞ ¼ OðDx6Þ; ð29aÞ

ð3xðþÞ1 � 3xð�Þ

1 � xðþÞ2 þ xð�Þ

2 þ xðþÞ3 � xð�Þ

3 Þ ¼ OðDx3Þ ð29bÞ
and
ðxk � �xkÞ ¼ OðDx2Þ: ð29cÞ

Here Eqs. (29a) and (29c) constrain both the ‘‘+’’ and ‘‘�’’ stencil, and thus the superscript (±) has been

dropped.Note that the non-oscillatory weights developed in [1] are formulated by appealing to Eqs. (29a) and

(29c) only (cf.P2

k¼0xk ¼ 1 and Eq. (2.14) with r = 3 in [1, p. 205]). Eq. (29b) is not presented there and thus

the constraints presented in [1] are incomplete. Comparison of Eqs. (25) and (29) reveals that the relaxation

of Eq. (25b) by one order of accuracy requires that Eq. (29b) be satisfied as well. Unfortunately, Eq. (29b) is

a difficult constraint to use in the design of the non-oscillatory weights.

4. The non-oscillatory weights of [1]

The development of fifth-order WENO schemes up to this point has been general, culminating in Eqs.

(25) and (29). All WENO schemes employing the candidate stencils in Fig. 3 retaining fifth-order conver-

gence properties must satisfy Eq. (29). The weights as designed in [1] are the focus of this section. This

scheme is denoted WENO5 in order to conform with the literature to date.


4.1. Formulation of x(JS)k

In an attempt to satisfy these constraints, weights were formulated in [1] according to

xðJSÞk ¼ akP2

i¼0ai; where ak ¼

�xk

ð�þ bkÞp ; ð30Þ

and a superscript (JS) is given to xk to denote the non-oscillatory weights as developed by Jiang and Shu

[1]. Here, � is a small parameter chosen such that ak remains bounded, bk is the indicator of smoothness for

the f̂k

jþ1=2 approximation of the numerical flux, and p is chosen in order to allow for weights in non-smooth

regions to approach zero at an accelerated rate as Dx ! 0. In much of the literature, the indicator of

smoothness is denoted by IS. It is easy to see that Eq. (29a) is satisfied sinceP2

k¼0xðJSÞk ¼

P2

k¼0 �xk ¼ 1

by construction.The indicators of smoothness as incorporated in Eq. (30) will provide a measure of the smoothness of a

solution over a particular stencil. Furthermore, the bk�s must be chosen such that each xðJSÞk approaches the

ideal weight at a fast enough convergence rate, according to Eq. (29c). For the present analysis, � is assumed

to be 0. The role of � will be analyzed in Section 4.3.

It is assumed that the indicator of smoothness can be written as

bk ¼ Dð1þOðDxÞ2Þ; ð31Þ

where D is some non-zero constant independent of k. Substitution of Eq. (31) into Eq. (30) gives

ak ¼�xk

ðDð1þOðDxÞ2ÞÞp¼ �xk

Dp ð1þOðDx2ÞÞ: ð32Þ

The sum of these terms is given by

X2k¼0

ak ¼1

Dp ð1þOðDx2ÞÞ ð33Þ

taking into account thatP2

k¼0 �xk ¼ 1. Thus xðJSÞk , as defined in Eq. (30), is given by

xðJSÞk ¼ �xk þOðDx2Þ; ð34Þ

which agrees with Eq. (29c). Note that Eq. (29b) is still not satisfied by enforcing Eq. (31).

4.2. Indicators of smoothness

It remains to find a set of bk which satisfy Eq. (31). Following [1], the indicators of smoothness are de-

fined by

bk ¼X2i¼1

Dx2i�1

Z xjþ1=2

xj�1=2

dif̂k

dxi

!2

dx: ð35Þ

Since f̂k ¼ b0 þ b1xþ b2x2, this equation simplifies to

bk ¼ b21Dx2 þ 13b22Dx

4

3; ð36Þ

where the stencils are centered around xj = 0. Substituting b1 and b2 from Eq. (14), the indicators ofsmoothness take on a particularly intuitive form:

Table

Orders

f 0 6¼ 0

b0 = (f

b1 = (f

b2 = (f


b0 ¼13

12ðfj�2 � 2f j�1 þ fj|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Dx2ðf 00j þOðDxÞÞ

Þ2 þ 1

4ðfj�2 � 4f j�1 þ 3f j|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2Dxðf 0jþOðDx2ÞÞ

Þ2;

b1 ¼13

12ðfj�1 � 2f j þ fjþ1|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Dx2ðf 00j þOðDx2ÞÞ

Þ2 þ 1

4ðfjþ1 � fj�1|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}2Dxf 0jþOðDx2Þ

Þ2;

b2 ¼13

12ðfj � 2f jþ1 þ fjþ2|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Dx2ðf 00j þOðDxÞÞ

Þ2 þ 1

4ð3f j � 4f jþ1 þ fjþ2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

�2Dxðf 0jþOðDx2ÞÞ

Þ2:

ð37Þ

Expansion of Eq. (37) in Taylor series about fj gives

b0 ¼ f 02Dx2 þ 13

12f 002 � 2

3f 0f 000

� �Dx4 þ � 13

6f 00f 000 þ 1

2f 0f ð4Þ

� �Dx5 þOðDx6Þ;

b1 ¼ f 02Dx2 þ 13

12f 002 þ 1

3f 0f 000

� �Dx4 þOðDx6Þ;

b2 ¼ f 02Dx2 þ 13

12f 002 � 2

3f 0f 000

� �Dx4 þ 13

6f 00f 000 � 1

2f 0f ð4Þ

� �Dx5 þOðDx6Þ:

ð38Þ

Referring to Table 1, the bk are of the form Eq. (31) for f 0 6¼ 0.

However, for the case that f 0 = O(Dx), the indicators of smoothness, b0 and b2, do not satisfy the con-

straint given in Eq. (31). Thus, xðJSÞk does not approach �xk at a fast enough rate of convergence to achieve

fifth-order accuracy near critical points. Rather,

xðJSÞk ¼ �xk þOðDxÞ ð39Þ

near critical points. One should note the discrepancy between the b0 and b2 in Table 1 and those reported in[1, Eqs. (3.5), (3.7) and (3.9)].

To demonstrate this discrepancy, one must consider an example in which f 0 = 0 at a point for which

f000 6¼0. Note that the example functions used in [1], namely sin(x) and sin4(x), do not satisfy this criterion

since they give f 0 = f000 = 0 at critical points. Due to this fact, as well as the introduction of � into the weights,

the convergence test of [1] is unable to detect this potential loss of accuracy at critical points.

4.3. The role of �

In order to understand the convergence behavior of WENO5 in much of the literature, a better under-

standing of the parameter � in Eq. (30) is required.

It should first be noticed that � appears as a dimensional quantity in Eq. (30) [11]. Thus � is normally

selected on a case by case basis, which belies the fact that � changes the order of convergence and is not

simply selected so as to prevent an indeterminate form in Eq. (30).

1

of bk written for comparison with Eq. (31)

f 0 = 0

0Dx)2(1 + O(Dx2)) b0 ¼ 1312ðf 00Dx2Þ2ð1þOðDxÞÞ

0Dx)2(1 + O(Dx2)) b1 ¼ 1312ðf 00Dx2Þ2ð1þOðDx2ÞÞ

0Dx)2(1 + O(Dx2)) b2 ¼ 1312ðf 00Dx2Þ2ð1þOðDxÞÞ


Substitution of Eq. (38) into Eq. (30) leads to

xðJSÞk ¼ akP2

i¼0ai; ð40aÞ

where

ak ¼�xk

�þ f 02Dx2 þOðDx4Þ : ð40bÞ

As Dx ! 0 in smooth regions of flow, the denominator of Eq. (40b) is eventually dominated by �, and the

indicators of smoothness become inconsequential. Of course, the magnitude of � determines at what reso-lution � becomes the dominant term. As � begins to dominate, the predictions of the WENO5 scheme

approach those of the central difference scheme, as is verified by considering Eq. (30) with bk = 0. Further-

more, oscillations of order �2 can exist near discontinuities, so choosing an � which is too large mitigates the

ENO behavior of the method.

The choice of � should also depend on the particular machine performing the calculations. To avoid divi-

sion by zero, � can be slightly larger than the square root of the smallest positive number allowed for a par-

ticular machine. According to the IEEE standard [13], this roughly translates to � > 10�18 for 32-bit floating

point arithmetic (single precision), � > 10�153 for 64-bit floating point arithmetic (double precision) and� > 10�2467 for 128-bit floating point arithmetic (quadruple precision). For schemes beyond 11th order,

i.e., schemes with potentially very small numerators in Eq. (30), these weights may need to be increased

slightly to avoid computing outside the range of machine precision. In the following Sections 4.5, 5.1

and 5.3, 128-bit floating point arithmetic was utilized to ensure machine precision is not playing a role.

In Sections 5.4 and 5.5, only 64-bit floating point arithmetic was used, since the solutions involving

captured shocks are not so accurate as to necessitate the higher precision.

4.4. The WENO5 scheme

The final form of the WENO5 scheme as published in [1] is thus given by

dfdx

��xj

¼f̂ jþ1=2 � f̂ j�1=2

Dxð41aÞ

and

f̂ j�1=2 ¼X2k¼0

xðJSÞk f̂

k

j�1=2; ð41bÞ

where the weights in Eq. (12) are explicitly denoted as those developed in [1]. These weights are given in Eq.

(30), and the bk are given in Eq. (37).

4.5. A simple example

Measuring the convergence of the WENO5 method to f 0(0) = 0 for

f ðxÞ ¼ x3 þ cosðxÞ; ð42Þ

will demonstrate the expected loss in accuracy, since f 0(0) = 0 and f000(0) = 6. Also, the effect � has on the

achieved rate of convergence is illustrated and explained. The function and its first and third derivatives

are shown in Fig. 4.

Examining the indicators of smoothness while resolving Dx clarifies how a fixed � eventually becomes the

dominant factor in determining the non-oscillatory weights. In Table 2, Dx is resolved from 10�3 to

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

4

5

6

7

f (x)=6+sin(x)

x

f (x)=3 x -sin(x)2

f(x)=x +cos(x)3

′′′f,f

,f′′′′

′

Fig. 4. Flux function from Section 4.5. Also shown are the first and third spatial derivatives. Note there are two critical points, both of

which have a non-vanishing third spatial derivative.

Table 2

Indicators of smoothness for calculation of dfdx

��x¼0

, where f(x) = x3 + cos(x)

Dx b0 b1 b2

1.00 · 10�3 1.09638 · 10�12 1.08333 · 10�12 1.07038 · 10�12

2.11149 · 10�12 2.10439 · 10�12 2.08533 · 10�12

5.00 · 10�4 6.81152 · 10�14 6.77083 · 10�14 6.73027 · 10�14

1.31086 · 10�13 1.30865 · 10�13 1.30271 · 10�13

2.50 · 10�4 4.24448 · 10�15 4.23177 · 10�15 4.21909 · 10�15

8.16540 · 10�15 8.15854 · 10�15 8.13997 · 10�15

1.25 · 10�4 2.64883 · 10�16 2.64486 · 10�16 2.64089 · 10�16

5.09481 · 10�16 5.09267 · 10�16 5.08687 · 10�16

6.25 · 10�5 1.65428 · 10�17 1.65304 · 10�17 1.65180 · 10�17

3.18159 · 10�17 3.18092 · 10�17 3.17911 · 10�17

For each successive value of Dx, the first row of indicators shown are used in calculating f̂þ1=2 while the second are used for calculating

f̂�1=2.


6.25 · 10�5 by factors of two. The indicators of smoothness approach zero at the rates given in Table 1 and

range from approximately 10�12 to approximately 10�17 as the grid is resolved. Thus for � 2 [10�12, 10�17],

� will change the order of convergence of the scheme as the grid is resolved, since it will begin to dominate

the bk at some point in the grid convergence process.

To demonstrate this behavior, results are compared for � 2 {10�40, 10�15,10�6} with one another and

with the results from the central scheme. Table 3 shows the error in approximating of/ox for

f(x) = x3 + cos(x) at x = 0. Also shown are the rates of convergence, rc, of each scheme. For � = 10�40,

the WENO5 scheme is confirmed to be third-order accurate at the critical point. As the size of � is increased

Table 3

Convergence properties of WENO5 as implemented in [1] near a critical point

WENO5 as implemented in [1] Upstream central scheme

� Dx rc Error Dx rc Error

10�40 1.00 · 10�3 – 1.63079 · 10�9 1.00 · 10�3 – 1.66667 · 10�17

5.00 · 10�4 2.95774 2.09907 · 10�10 5.00 · 10�4 5.00000 5.20833 · 10�19

2.50 · 10�4 2.97924 2.66187 · 10�11 2.50 · 10�4 5.00000 1.62760 · 10�20

1.25 · 10�4 2.98971 3.35115 · 10�12 1.25 · 10�4 5.00000 5.08626 · 10�22

6.25 · 10�5 2.99488 4.20384 · 10�13 6.25 · 10�5 5.00000 1.58946 · 10�23

10�15 1.00 · 10�3 – 1.63016 · 10�9

5.00 · 10�4 2.96716 2.08462 · 10�10

2.50 · 10�4 3.13003 2.38119 · 10�11

1.25 · 10�4 4.38773 1.13751 · 10�12

6.25 · 10�5 6.45124 1.29998 · 10�14

10�6 1.00 · 10�3 – 3.62634 · 10�15

5.00 · 10�4 6.98214 2.86836 · 10�17

2.50 · 10�4 6.92438 2.36150 · 10�19

1.25 · 10�4 6.72923 2.22582 · 10�21

6.25 · 10�5 6.24690 2.93079 · 10�23


to 10�15, this parameter begins to dominate the indicator of smoothness. The use of � = 10�6 is shown, since

it is recommended in the literature [1]. For this case, the method is ostensibly even seventh-order convergent

at the first level of resolution, dropping to lower orders as the method becomes almost equivalent to the

central scheme.

It is clear that the order of the WENO5 scheme is dependent on the size of � and the level of grid res-

olution. Thus it becomes difficult, if not impossible, to determine the order of the WENO5 scheme for a

complicated problem a priori at a given grid resolution.

One may inquire into the behavior of fifth-order WENO schemes at critical points of higher order (i.e., f 0

and f 00 are zero). To answer this, consider that the indicators of smoothness, as given by Eqs. (36) and (14),

are linear combinations of derivatives evaluated at the stencil midpoint. In the case where the component

stencils are third-order polynomials, only the first and and second derivatives can be approximated. When

both of these are zero at the midpoint of the stencil, the indicator of smoothness will no longer approximate

a non-zero quantity. In other words, the indicators of smoothness are determined to the lowest order by

truncation error. Thus Eq. (31) is not satisfied, since such an error will, in general, not admit a single lowest

order term, D, for all the bk. This results in a loss of accuracy.

5. Fifth-order mapped WENO (WENO5M)

Formulation of the weights as introduced in [1] is both intuitive and fifth-order accurate for smooth

flows except near critical points. Thus, rather than attempt to formulate a new indicator of smoothness

which allows our weights to satisfy Eq. (29), the xðJSÞk are used as a first approximation and mapped to more

accurate values so that Eq. (25) is satisfied.

5.1. The correction

To increase the accuracy of these weights, consider the functions

gkðxÞ ¼xð�xk þ �x2

k � 3�xkxþ x2Þ�x2k þ xð1� 2�xkÞ

; �xk 2 ð0; 1Þ ð43Þ


for k = 0, 1, 2. All of these functions have the following features: they are monotonically increasing with finite

slope, gkð0Þ ¼ 0; gkð1Þ ¼ 1; gkð�xkÞ ¼ �xk; g0kð�xkÞ ¼ 0 and g00kð�xkÞ ¼ 0. These functions are shown in Fig. 5.

A more accurate approximation of the weights is given by

Fig. 5

identifi

an x(J

a�k ¼ gkðxðJSÞk Þ: ð44Þ

To understand this process, it is instructive to view each gk(x) as a mapping. For each k, gk(x) becomes flat

in the neighborhood of the kth ideal weight. Thus xðJSÞk , which is in an O(Dx) neighborhood of �xk according

to Eq. (39), can be used as an initial guess which is mapped to a more accurate value by gk(x). For values ofxðJSÞ

k which are close to 0 or 1, the mapping has a minimal effect. This behavior can be seen geometrically

from inspection of Fig. 5 and consideration of the identity mapping.

Evaluation at xðJSÞk of the Taylor series approximations of the gk(x) about �xk yield

a�k ¼ gkð�xkÞ þ g0kð�xkÞ xðJSÞk � �xk

� �þ g00kð�xkÞ

2xðJSÞ

k � �xk

� �2þ g000k ð�xkÞ

6xðJSÞ

k � �xk

� �3þ � � �

¼ �xk þxðJSÞ

k � �xk

� �3�xk � �x3

k

þ � � �

¼ �xk þOðDx3Þ:

ð45Þ

The modified weights are defined according to

xðMÞk ¼ a�kP2

i¼0a�i

where1P2

i¼0a�i

¼ 1þOðDx3Þ; ð46Þ

again taking into account thatP2

k¼0 �xk ¼ 1. Here a superscript (M) has been added to signify the weights

use in WENO5M. Thus,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k = 0

k = 2

k = 1

ω

α* k

Identity mapping

. Mapping functions used with the modified WENO5 scheme given in Section 5. Notice the identity mapping, also shown,

es the stationary points for the a�k maps. Each a�k becomes flat in the region around the corresponding �xk , which effectively mapsS) in this region to a more accurate approximation of the ideal weight.

Table

Conve

WENO

Dx

1.00 ·5.00 ·2.50 ·1.25 ·6.25 ·


X2k¼0

xðMÞk ¼ 1 and xðMÞ

k ¼ �xk þOðDx3Þ ð47Þ

which satisfies Eq. (25). Thus the method is fifth-order accurate even near critical points where f 0 = 0. Note

that implementation of this correction requires only a slight alteration to the existing WENO5 interpolation

scheme.

For comparison, the results using the mapped fifth-order WENO (WENO5M) scheme on the example

problem given in Section 4.5 are given in Table 4. The method is shown to be fifth-order accurate with

� = 10�40.

5.2. The WENO5M scheme

The final form of the WENO5M scheme is thus given by

dfdx

��xj

¼f̂ jþ1=2 � f̂ j�1=2

Dxð48aÞ

and

f̂ j�1=2 ¼X2k¼0

xðMÞk f̂

k

j�1=2; ð48bÞ

where the weights in Eq. (12) are explicitly denoted as the mapped weights. These weights are given in Eq.

(46) where a�k ¼ gkðxðJSÞk Þ. Here, xðJSÞ

k is given in Eqs. (30) and (37).

5.3. Linear advection example

As an example, consider

ouot

þ ouox

¼ 0 on x 2 ½�1; 1�; t 2 ½0; 2� ð49Þ

with the initial condition

uðx; t ¼ 0Þ ¼ sin px� sinðpxÞp

� �ð50Þ

and periodic boundary conditions, u|x = �1 = u|x = 1. Here x0 = �1, xN = 1, and the flux f is simply the con-

served quantity itself, u. This particular initial condition has two critical points at which f 0 = 0 and f000 6¼ 0 as

seen in Fig. 6.

4

rgence properties of WENO5M

5M scheme (� = 10�40) Upstream central scheme

rc Error Dx rc Error

10�3 – 6.14598 · 10�14 1.00 · 10�3 – 1.66667 · 10�17

10�4 4.86269 2.11240 · 10�15 5.00 · 10�4 5.00000 5.20833 · 10�19

10�4 4.93600 6.90069 · 10�17 2.50 · 10�4 5.00000 1.62760 · 10�20

10�4 4.96904 2.20324 · 10�18 1.25 · 10�4 5.00000 5.08626 · 10�22

10�5 4.98477 6.95818 · 10�20 6.25 · 10�5 5.00000 1.58946 · 10�23

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

x

u(x,t=0)

u´(x,t=0)/ π

u,u´

/π,u

´´´/

π4

u´´´(x,t=0)/π4

Fig. 6. The initial condition, uðx; t ¼ 0Þ ¼ sinðpx� sinðpxÞp Þ, used for the example problem given in Section 5.3. Also shown are the first

and third derivatives. Note that there are two critical points which all have a non-vanishing third derivative.


The exact solution to Eqs. (49) and (50) is easily found using the method of characteristics and is given

by

uexactðx; tÞ ¼ sin pðx� tÞ � sinðpðx� tÞÞp

� �: ð51Þ

Spatial discretization of ou/ox in Eq. (49) is done using the upstream central scheme, the WENO5M scheme

with � = 10�40, and the WENO5 scheme with � = 10�6 and 10�40. For each case, the resulting set of ordin-ary differential equations in time are solved discretely using a third-order Runge–Kutta method given by

u� ¼ un þ DtLðunÞ;u�� ¼ 3

4un þ 1

4u� þ 1

4DtLðu�Þ;

unþ1 ¼ 13un þ 2

3u�� þ 2

3DtLðu��Þ;

ð52Þ

where L ¼ � f̂ jþ1=2�f̂ j�1=2

Dx and un represents the solution at time step n. Since this time integration method

incurs O(Dt3) errors, the time step is chosen to be Dt = 8Dx5/3 in order that the error for the overall schemeis a measure of the spatial convergence only.

The norm of the error is computed by comparison with the exact solution at time t = 2 according to

L1 ¼2

Nx

XNx

i¼0

jui � uexact;ij; ð53aÞ

L2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Nx

XNx

i¼0

ðui � uexact;iÞ2vuut ; ð53bÞ

L1 ¼ max jui � uexact;ij for i ¼ 0; . . . ;Nx; ð53cÞ


where Nx + 1 is the number of grid points in the domain [�1, 1]. Note that Nx = 2/Dx. These errors for var-ious values of Dx are given in Tables 5–8 as well as the order of convergence between each level of

resolution.

Table 5 shows the results for the upstream central scheme. This scheme, which gives optimal results for a

five point stencil, is seen to converge at fifth order. Comparison of the errors given in Tables 5 and 6 illus-trates how well the WENO5M scheme is able to mimic the upstream central scheme, even in the presence of

critical points where f000 6¼ 0 and where � is small. The WENO5M scheme is also seen to converge at fifth

order.

Table 5

Convergence properties of upstream central scheme

Dx L1 L2 L1

rc Norm rc Norm rc Norm

4.00 · 10�2 – 1.45316 · 10 �3 – 1.29424 · 10�3 – 1.80818 · 10�3

2.00 · 10�2 5.02392 4.46646 · 10�5 5.00981 4.01709 · 10�5 4.99935 5.65313 · 10�5

1.00 · 10�2 5.01324 1.38302 · 10�6 5.01072 1.24605 · 10�6 5.01194 1.75205 · 10�6

5.00 · 10�3 5.00261 4.31411 · 10�8 5.00225 3.88783 · 10�8 5.00254 5.46551 · 10�8

2.50 · 10�3 5.00179 1.34649 · 10�9 5.00173 1.21349 · 10�9 5.00193 1.70569 · 10�9

1.25 · 10�3 5.00053 4.20624 · 10�11 5.00052 3.79078 · 10�11 5.00062 5.32799 · 10�11

6.25 · 10�4 5.00010 1.31436 · 10�12 5.00010 1.18454 · 10�12 5.00014 1.66483 · 10�12

3.12 · 10�4 5.00000 4.10738 · 10�14 5.00000 3.70169 · 10�14 5.00002 5.20253 · 10�14

Table 6

Convergence properties of WENO5M (� = 10�40)

Dx L1 L2 L1


4.00 · 10�2 – 1.45252 · 10�3 – 1.29382 · 10�3 – 1.80832 · 10�3

2.00 · 10�2 5.02322 4.46665 · 10�5 5.00941 4.01691 · 10�5 4.99951 5.65292 · 10�5

1.00 · 10�2 5.01330 1.38302 · 10�6 5.01066 1.24604 · 10�6 5.01188 1.75205 · 10�6

5.00 · 10�3 5.00261 4.31411 · 10�8 5.00224 3.88783 · 10�8 5.00255 5.46551 · 10�8

2.50 · 10�3 5.00179 1.34649 · 10�9 5.00173 1.21349 · 10�9 5.00193 1.70569 · 10�9

1.25 · 10�3 5.00053 4.20624 · 10�11 5.00052 3.79078 · 10�11 5.00062 5.32799 · 10�11

6.25 · 10�4 5.00010 1.31436 · 10�12 5.00010 1.18454 · 10�12 5.00014 1.66483 · 10�12

3.12 · 10�4 5.00000 4.10738 · 10�14 5.00000 3.70169 · 10�14 5.00002 5.20253 · 10�14

Table 7

Convergence properties of WENO5 (� = 10�6) as implemented in [1]

Dx L1 L2 L1


4.00 · 10�2 – 1.57063 · 10�3 – 1.36967 · 10�3 – 1.96666 · 10�3

2.00 · 10�2 4.93920 5.11949 · 10�5 4.93272 4.48456 · 10�5 5.00709 6.11565 · 10�5

1.00 · 10�2 4.94455 1.66252 · 10�6 4.84158 1.56408 · 10�6 4.02856 3.74736 · 10�6

5.00 · 10�3 5.03706 5.06362 · 10�8 4.99207 4.91468 · 10�8 4.72751 1.41450 · 10�7

2.50 · 10�3 5.06115 1.51671 · 10�9 5.18207 1.35375 · 10�9 6.12173 2.03132 · 10�9

1.25 · 10�3 5.04054 4.60838 · 10�11 5.05252 4.07921 · 10�11 5.14205 5.75264 · 10�11

6.25 · 10�4 5.04409 1.39677 · 10�12 5.02493 1.25292 · 10�12 5.01446 1.77977 · 10�12

3.12 · 10�4 5.03707 4.25417 · 10�14 5.02796 3.84020 · 10�14 5.03535 5.42717 · 10�14

Table 8

Convergence properties of WENO5 (� = 10�40) as implemented in [1]

Dx L1 L2 L1


4.00 · 10�2 – 1.57063 · 10�3 – 1.36964 · 10�3 – 1.96669 · 10�3

2.00 · 10�2 4.93832 5.12262 · 10�5 4.93107 4.48960 · 10�5 5.00736 6.11465 · 10�5

1.00 · 10�2 4.90103 1.71450 · 10�6 4.77077 1.64461 · 10�6 3.81337 4.34944 · 10�6

5.00 · 10�3 4.70139 6.58987 · 10�8 4.28898 8.41301 · 10�8 3.39847 4.12469 · 10�7

2.50 · 10�3 4.57542 2.76401 · 10�9 3.84598 5.85055 · 10�9 3.29195 4.21128 · 10�8

1.25 · 10�3 4.42426 1.28737 · 10�10 3.70724 4.47926 · 10�10 3.26545 4.37942 · 10�9

6.25 · 10�4 4.30740 6.50202 · 10�12 3.68715 3.47748 · 10�11 3.26639 4.55131 · 10�10

3.12 · 10�4 4.22130 3.48587 · 10�13 3.68948 2.69539 · 10�12 3.27135 4.71371 · 10�11


Comparison of the results reported in Table 5 with those of Tables 7 and 8 also reveals some interesting

features of the WENO5 scheme. Table 7 shows that the order of convergence for � = 10�6 fluctuates around

fifth order. The error incurred at this level of grid refinement by using WENO5 with � = 10�40 is much

worse; in fact, the error is third order in the L1 norm as seen in Table 8.

The overall L1 convergence behavior of these schemes is seen in Fig. 7. Predictions given by the central

and WENO5M schemes are indistinguishable at the scale shown. Both of these solution sets converge at

fifth order as evidenced by the slope of these two lines. For WENO5, the convergence rate is seen to vary

with � and Dx. WENO5 with � = 10�16 is seen to converge consistently at third order. This third-order con-vergence line along with the fifth-order convergence line of the upstream central scheme form an envelope

which contains the other WENO5 cases. All the other WENO5 cases are seen to peal away from the third-

order line and asymptotically approach the fifth-order upstream central case. This is due to the gradual

domination of � over the bk as Dx is decreased. Note that the slope of the various WENO5 schemes are

10-3

10-2

10-12

10-10

10-8

10-6

10-4

∆x

L ∞ e

rror

Upstream central

WENO5M

ε = 10-40

WENO5

ε = 10-6

WENO5

ε = 10-8

WENO5

ε = 10-10

WENO5

ε = 10-16

O(∆x5) slope

O(∆x3) slope

Fig. 7. Convergence plot for (ou/ot + ou/ox) = 0 with the initial condition uðx; t ¼ 0Þ ¼ sinðpx� sinðpxÞp Þ for x 2 [�1,1] with periodic

boundary conditions. In each case Dt = 8Dx5/3.


seen to be greater than fifth order as � begins to dominate. This explains the ‘‘super-convergence’’ phenom-

enon in which the WENO5 schemes converge at higher than fifth order for some grid resolutions. Even so,

these schemes are never more accurate than the fifth-order upstream central scheme.

5.4. Euler example I: shock entropy wave interaction

For the one-dimensional Euler equations of gas dynamics, we have the conservation of mass, momentum

and energy:

oqot

þ oðquÞox

¼ 0;

oðquÞot

þ oðqu2 þ pÞox

¼ 0;

oEot

þ oðuE þ upÞox

¼ 0;

ð54Þ

where q, u, p and E are the density, particle velocity, pressure and total energy, respectively. The system of

equations is closed by specifying an equation of state; here, the calorically perfect ideal gas equation of state

is used:

E ¼ pðc� 1Þ þ 12qu2: ð55Þ

For the shock entropy wave interaction problem [7], c = 1.4 and the initial conditions are set by a Mach 3

shock interacting with a perturbed density field:

ðq; u; pÞ ¼ ð277; 4ffiffiffiffi35

p

9; 313Þ if x < �4;

ð1þ 15sin 5x; 0; 1Þ if x P �4

(ð56Þ

with zero-gradient boundary conditions at x = ±5.

The Roe scheme is used for calculating the fluxes in the system of hyperbolic conservation laws [7] andthe third-order Runge–Kutta scheme of Eq. (52) is used to advance in time. Other schemes, such as non-

decomposition based Lax–Friedrichs schemes [14,15], have been utilized with similar results. The solutions

at t = 1.8 for the Roe scheme are given in Figs. 8 and 9 for the WENO5 (� = 10�6) and WENO5M

(� = 10�40) schemes, respectively. Each was computed using Nx = Nt = 400, where Nt is the number of time

steps. It is clear, even though the formal rate of convergence is no better than first order due to the captured

shocks, that the WENO5M scheme resolves the salient features of the flow with higher fidelity. This is

particularly true at points in the flow where the solution�s first derivative vanishes.

5.5. Euler example II: interacting blast waves

The interacting blast wave example [16] is another difficult test of shock capturing schemes. The Euler

equation (55) with c = 1.4 are solved with initial conditions

ðq; u; pÞ ¼ð1; 0; 1000Þ if 0 6 x < 0:1;

ð1; 0; 0:01Þ if 0:1 6 x < 0:9;

ð1; 0; 100Þ if 0:9 6 x 6 1;

8><>: ð57Þ

with reflection boundary conditions at x = 0 and x = 1.

Again, the Roe scheme and third-order Runge–Kutta are used in constructing the numerical solution.

The solutions at t = 0.038 are given in Figs. 10 and 11 for the WENO5 (� = 10�6) and WENO5M

Fig. 9. Shock entropy wave interaction. Density plot of WENO5M (� = 10�40) solution at t = 1.8 with Nx = Nt = 400, e, and well-

resolved WENO5M (� = 10�40) solution with Nx = Nt = 12,800, solid line.

Fig. 8. Shock entropy wave interaction. Density plot of WENO5 (� = 10�6) solution at t = 1.8 with Nx = Nt = 400, e, and well-

resolved WENO5M (� = 10�40) solution with Nx = Nt = 12,800, solid line.


(� = 10�40) schemes, respectively. Each was computed using Nx = 400 and Nt = 800. Careful examination

reveals that the WENO5M scheme resolves the salient features of the flow with higher fidelity. The three

contact waves, near x = 0.595, 0.765, 0.799, are better resolved as too are the various peaks and valleys.

Fig. 10. Interacting blast waves. Density plot of WENO5 (� = 10�6) solution at t = 0.038 with Nx = 400, Nt = 800,e, and well-resolved

WENO5M (� = 10�40) solution with Nx = 12,800, Nt = 25,600, solid line.

Fig. 11. Interacting blast waves. Density plot of WENO5M (� = 10�40) solution at t = 0.038 with Nx = 400, Nt = 800, e, and well-

resolved WENO5M (� = 10�40) solution with Nx = 12,800, Nt = 25,600, solid line.



5.6. Euler example III: Riemann problem of lax

The Riemann problem of Lax [17] was simulated with both the WENO5 and WENO5M methods. The

initial conditions are:

Fig. 12

(� = 10

ðq; u; pÞ ¼ð0:445; 0:698; 3:528Þ if x 6 0;

ð0:5; 0; 0:571Þ if x > 0

ð58Þ

with zero gradient boundary conditions at x = ±0.5. The solution was integrated to t = 0.13 with the Roe

scheme and third-order Runge–Kutta. For Nt = Nx = 100, Fig. 12 demonstrates the improved resolution

ability of the present method, while still maintaining the ENO property.

5.7. Euler example IV: Riemann problem of sod

Lastly, the standard shock tube problem of Sod [12] was simulated with both the WENO5 and WE-

NO5Mmethods, and the grid was refined so as to enable determination of the convergence rates. The initial

conditions are:

ðq; u; pÞ ¼ð1; 0; 1Þ if x 6 0;

ð0:125; 0; 0:1Þ if x > 0

ð59Þ

with zero gradient boundary conditions at x = ±0.5. The solution was integrated to t = 0.14385 with the

Roe scheme and third-order Runge–Kutta. The results of the convergence study, with Nt = Nx, are summa-

rized in Table 9. Both methods predict qualitatively similar results, but the WENO5M method consistently

predicts solutions with roughly 10% less error than the WENO5 method, similar to Fig. 12. The rateof convergence of both are roughly 5/6 order as Dx ! 0. This convergence rate, at or below first order,

. Riemann problem of Lax [17]. Density plot of WENO5 (� = 10�6) solution at t = 0.13 with Nx = Nt = 100, e; WENO5M�40) solution with Nx = Nt = 100, s; Exact solution, solid line.

Table 9

L1 density error norms and convergence rates, rc, for WENO5 (� = 10�6) as implemented in Ref. [1] and WENO5M (� = 10�40)

methods applied to Sod�s [12] shock tube problem

Dx L1 rc

WENO5 WENO5M WENO5 WENO5M

1/100 6.88 · 10�3 6.35 · 10�3 – –

1/200 3.67 · 10�3 3.34 · 10�3 0.907 0.925

1/400 2.01 · 10�3 1.84 · 10�3 0.866 0.860

1/800 1.03 · 10�3 9.37 · 10�4 0.970 0.975

1/1600 5.11 · 10�4 4.59 · 10�4 1.008 1.030

1/3200 2.76 · 10�4 2.48 · 10�4 0.889 0.885

1/6400 1.38 · 10�4 1.23 · 10�4 0.998 1.018

1/12800 7.27 · 10�5 6.40 · 10�5 0.927 0.938

1/25600 3.89 · 10�5 3.42 · 10�5 0.900 0.906

1/51200 2.18 · 10�5 1.91 · 10�5 0.837 0.837

1/102400 1.20 · 10�5 1.09 · 10�5 0.858 0.817


is characteristic of all shock capturing schemes when applied to flows with embedded discontinuities. In

particular, nominally fifth-order shock capturing algorithms converge at roughly 5/6 order in the L1 norm

for discontinuities in linearly degenerate fields [18]. This is commensurate with the notion that a captured

linear discontinuity will smear over a number of computational zones proportional to N 1=ðrcþ1Þx , where rc is

the nominal order of the scheme [19].

6. Summary of WENO and mapped WENO schemes of 3rd to 11th order

WENO schemes (and their monotonicity-preserving counterparts) up to 11th order have been outlined

in [9]. Notationally, let nCP denote the order of the critical point. For example nCP = 0 corresponds to

f 0 6¼ 0; nCP = 1 corresponds to f 0 = 0, f 00 6¼ 0; nCP = 2 corresponds to f 0 = 0, f 00 = 0, f 000 6¼ 0, etc.

As with the fifth-order WENO scheme, the detailed Taylor series analysis of these higher order WENO

schemes can be performed, under the assumption � = 0. It is easily verified that away from critical points,

the schemes are (2r � 1)th order accurate as advertised, but all of the schemes drop by 2 orders at points

near nCP = 1. For each further increase in nCP, the schemes continue to degrade by an order of accuracy,until the schemes are only (r � 1)th order accurate. Inferring from the results, it is found that the rate of

convergence can be related to r and nCP in the following manner (at least for r 6 6):

rc ¼2r � 1 if not near a critical point;

maxð2r � 2� nCP; nCPÞ otherwise:

ð60Þ

Note that once nCP > r � 1, the rate of convergence actually starts to increase. This is simply due to the fact

that the function is so flat that any consistent scheme will yield better convergence locally (for example, a

first-order scheme, r = 1, is actually second-order accurate at x = 0 for the cubic function f(x) = x3, i.e.

nCP = 2).

Table 10 demonstrates the convergence rates for the WENO schemes of r = 2, 3, 4, 5, 6, corresponding to

Eq. (60). Also shown are the rates of convergence for the mapped WENO schemes, with either 1 or 2 appli-

cations of Eq. (44) followed by Eq. (46). Note that for nCP = r � 1, the individual weights are O(1) away

from being the ideal weights, and thus the mappings are ineffective. At these points, it is difficult for anyscheme to distinguish the difference between a discontinuity and a function that is nearly flat but relatively

rapidly changing on the grid. For the cases where 1 6 nCP < r � 1, it is predicted that the mapped WENO

schemes can recover optimal convergence rates.

Table 10

Rates of convergence, rc, for WENO and mapped WENO schemes as a function of critical point order

nCP rc-WENO rc-WENO-1 mapping rc-WENO-2 mappings

r = 2 0 3 3 3

1 1 1 1

r = 3 0 5 5 5

1 3 5 5

2 2 2 2

r = 4 0 7 7 7

1 5 7 7

2 4 6 7

3 3 3 3

r = 5 0 9 9 9

1 7 9 9

2 6 9 9

3 5 7 9

4 4 4 4

r = 6 0 11 11 11

1 9 11 11

2 8 11 11

3 7 11 11

4 6 8 11

5 5 5 5


7. Conclusion

It has been shown that the previously published constraints for fifth-order accurate WENO schemes are

incomplete. The extensions to higher orders are similarly afflicted near critical points. Former algorithms

for finding these weights do not satisfy this incomplete set of criteria in the neighborhood of critical points

and, in the worst case, result in (r � 1)th order convergence for smooth flows. Furthermore, an explanation

has been given of the dramatic effect that �, in conjunction with the level of grid resolution, has on the con-

vergence of WENO5 schemes. A correction is given which achieves the full (2r � 1)th order accuracy in theneighborhood of critical points where f 0 = 0, even for very small �.

Lastly, while it is clear that the improvement described here allows true fifth-order convergence rates and

consequent high accuracy to be realized when approximating smooth solutions, the improvement in accu-

racy is more modest when flows with embedded discontinuities are considered. Certainly for such flows,

side by side comparisons reveal that the predictions of WENO5M are better than those of WENO5. How-

ever, in both cases the rate of convergence dramatically drops to no more than first order. Consequently,

the improvement for flows with shocks is proportional rather than geometric. While this is often recognized

in much of the computational literature, it is as often unrecognized; the enthusiasm displayed by someadvocates of high order methods for problems with shocks should be proportionately tempered.

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